Maharashtra State BoardHSC Science (Electronics) 11th
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Prove the following identities: 1-secθ+tanθ1+secθ-tanθ=secθ+tanθ-1secθ+tanθ+1 - Mathematics and Statistics

Sum

Prove the following identities:

`(1 - sectheta + tan theta)/(1 + sec theta - tan theta) = (sectheta + tantheta - 1)/(sectheta + tantheta + 1)`

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Solution

We know that,

1 + tan2θ = sec2θ

∴ sec2θ – tan2θ = 1

∴ (secθ – tanθ)(secθ + tanθ) = 1·1

∴ `(sectheta + tantheta)/1 = 1/(sectheta - tantheta)`

By the theorem of equal roots,

`(sectheta + tantheta)/1 = 1/(sectheta - tantheta)`

= `(sectheta + tantheta + 1)/(1 + sectheta - tantheta)`

= `(sectheta + tantheta - 1)/(1 - sectheta + tantheta)`

∴ `(sectheta + tantheta + 1)/(1 + sectheta - tantheta) = (sectheta + tantheta - 1)/(1 - sectheta + tantheta)`

∴ `(1 - sectheta + tantheta)/(1 + sectheta - tantheta) = (sectheta + tantheta - 1)/(sectheta + tantheta + 1)`.

Concept: Fundamental Identities
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